Exponential Functions Section 3.1

What are Exponential Functions?

Why study exponential functions? Many real-life situations can be described using exponential functions, including – Population Growth – Growth of epidemics – Radioactive decay – Compound Interest

Definition of Exponential Function The exponential function f with base a is defined by f(x) = a x or y = a x where a is a positive number other and 1 (a>0 and a ≠ 1) and x is any real number.

Exponential Function f(x) = a x Domain: (-∞, ∞) Range (0, ∞) y-intercept: (0, 1) NO zero (has a horizontal asymptote at y=0) Increasing (-∞, ∞) No relative minimum or maximum Neither even nor odd Continuous Has an inverse (logarithm) Xy=f(x)

f(x) = a (bx-c) + d Transformations learned in Chapter 1 still apply Parent is exponential function with base a Vertical translation –”d” Horizontal translation –”bx-c=0” Reflection on x-axis – “sign of a” Reflection on y-axis-”sign of b” Vertical Stretch or Shrink – “numeric value of a” EXAMPLES

Applications

Example 1: You take out a loan of $30,000 to buy a new car. The bank loans you the money at 7.5% annual interest for 5 years compounded monthly.

Applications Example 2 You deposit $1 into an account paying 100% interest compounded: a)Yearly b) semiannually c) quarterly d) monthly e) weekly f) daily g) hourly h) by the minute i) by the second j) “continuously” PrnA 1112annually semiannually quarterly monthly weekly daily hourly minute second

“e” ---Natural number An irrational number (lots of decimal places) Denoted by “e” in honor of Leonard Euler As n→∞, the approximate value of “e” to nine decimal places is e ≈ …….

Applications

Example 3: You invest $5000 for 10 years at an interest rate of 6.5%. If continuous compounding occurs, how much money will you have in 10 years?

Applications

Assignment Page 396 #25-31 odd, odd, odd, odd, 73